If $A$,$B$,and $C$ are square matrices of order $3$ such that $A = \begin{bmatrix} x & 0 & 1 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ and $|B| = 36$,$|C| = 4$,$(x, y, z \in \mathbb{N})$ and $|ABC| = 1152$,then the minimum value of $x + y + z$ is

Let $\Omega$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either $0$ or $1$. Five of these entries are $1$ and four of them are $0$.
$1.$ The number of matrices in $\Omega$ is
$(A) 12$ $(B) 6$ $(C) 9$ $(D) 3$
$2.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has a unique solution,is
$(A)$ less than $4$ $(B)$ at least $4$ but less than $7$ $(C)$ at least $7$ but less than $10$ $(D)$ at least $10$
$3.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ is inconsistent,is
$(A) 0$ $(B)$ more than $2$ $(C) 2$ $(D) 1$

If $A = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & \sin^2 \beta \end{bmatrix}$ are two matrices such that the product $AB$ is a null matrix,then $\alpha - \beta$ is:

If $a, b, c$ and $d$ are complex numbers,then the determinant $\Delta = \begin{vmatrix} 2 & a+b+c+d & ab+cd \\ a+b+c+d & 2(a+b)(c+d) & ab(c+d)+cd(a+b) \\ ab+cd & ab(c+d)+cd(a+b) & 2abcd \end{vmatrix}$ is

If $a_{r} = \cos \frac{2 r \pi}{9} + i \sin \frac{2 r \pi}{9}$,$r = 1, 2, 3, \ldots$,$i = \sqrt{-1}$,then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to:

For $A = \begin{bmatrix} 0 & 0 & -2 \\ 0 & -2 & 0 \\ -2 & 0 & 0 \end{bmatrix}$,which of the following is true?